Similarity of figures is a property that is used to compare figures. Two triangles are said to be similar if their corresponding sides are in equal ratio and if their corresponding angles are equal. If the corresponding sides of two triangles are equal, then these triangles are called equiangular triangles. In an equiangular triangle, the ratio of the corresponding sides of the two sides is always the same. This property of triangles follows from the basic proportionality theorem.
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These NCERT solutions of Class 10 maths ex 6.2 contain two theorems: (Theorem 1) When a parallel line is drawn along one of the triangle's sides and intersects the other two sides at certain points, the two sides are divided in the same proportion. (Theorem 2) Any line that divides the two sides of a triangle in the same ratio is parallel to the third side. NCERT solutions for exercise 6.2 Class 10 Maths chapter 6 Triangles discuss the concept as per the NCERT Book, like a triangle as a polygon, the interior angles of a triangle, the exterior angles of a triangle, Pythagoras' theorem, and conditions of similarity in triangles.
Triangles Class 10 Chapter 6 Exercise: 6.2
Q1 In Fig. 6.17, (i) and (ii), DE || BC. Find EC in (i) and AD in (ii).
Answer:
(i)
Let EC be x
Given: DE || BC
By using the proportionality theorem, we get
$\frac{AD}{DB}=\frac{AE}{EC}$
$\Rightarrow \frac{1.5}{3}=\frac{1}{x}$
$\Rightarrow x=\frac{3}{1.5}=2\, cm$
$\therefore EC=2\, cm$
(ii)
Let AD be x
Given: DE || BC
By using the proportionality theorem, we get
$\frac{AD}{DB}=\frac{AE}{EC}$
$\Rightarrow \frac{x}{7.2}=\frac{1.8}{5.4}$
$\Rightarrow x=\frac{7.2}{3}=2.4\, cm$
$\therefore AD=2.4\, cm$
Answer:
(i)
Given :
PE = 3.9 cm, EQ = 3 cm, PF = 3.6 cm and FR = 2.4 cm
$\frac{PE}{EQ}=\frac{3.9}{3}=1.3\, cm$ and $\frac{PF}{FR}=\frac{3.6}{2.4}=1.5\, cm$
We have
$\frac{PE}{EQ} \neq \frac{PF}{FR}$
Hence, EF is not parallel to QR.
Answer:
(ii)
Given :
PE = 4 cm, QE = 4.5 cm, PF = 8 cm and RF = 9 cm
$\frac{PE}{EQ}=\frac{4}{4.5}=\frac{8}{9}\, cm$ and $\frac{PF}{FR}=\frac{8}{9}\, cm$
We have
$\frac{PE}{EQ} = \frac{PF}{FR}$
Hence, EF is parallel to QR.
Q2 (3) E and F are points on the sides PQ and PR respectively of a triangle PQR. For each of the following cases, state whether EF || QR : PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
Answer:
(iii)
Given:
PQ = 1.28 cm, PR = 2.56 cm, PE = 0.18 cm and PF = 0.36 cm
$\frac{PE}{PQ}=\frac{0.18}{1.28}=\frac{9}{64}\, cm$ and $\frac{PF}{PR}=\frac{0.36}{2.56}=\frac{9}{64}\, cm$
We have
$\frac{PE}{EQ} = \frac{PF}{FR}$
Hence, EF is parallel to QR.
Q3 In Fig. 6.18, if LM || CB and LN || CD, prove that $\frac{AM}{AB} = \frac{AN}{AD }$
Answer:
Given : LM || CB and LN || CD
To prove :
$\frac{AM}{AB} = \frac{AN}{AD }$
Since , LM || CB so we have
$\frac{AM}{AB}=\frac{AL}{AC}............(1)$
Also, LN || CD
$\frac{AL}{AC}=\frac{AN}{AD}..............(2)$
From equations 1 and 2, we have
$\frac{AM}{AB} = \frac{AN}{AD }$
Hence proved.
Q4 In Fig. 6.19, DE || AC and DF || AE. Prove that BF / FE = BE / EC
Answer:
Given: DE || AC and DF || AE.
To prove:
$\frac{BF}{FE} = \frac{BE}{EC }$
Since , DE || AC so we have
$\frac{BD}{DA}=\frac{BE}{EC}...........(1)$
Also, DF || AE
$\frac{BD}{DA}=\frac{BF}{FE}.............(2)$
From equations (1) and (2), we have
$\frac{BF}{FE} = \frac{BE}{EC }$
Hence proved.
Q5 In Fig. 6.20, DE || OQ and DF || OR. Show that EF || QR.
Answer:
Given: DE || OQ and DF || OR.
To prove EF || QR.
Since DE || OQ, we have
$\frac{PE}{EQ}=\frac{PD}{DO}..........(1)$
Also, DF || OR
$\frac{PF}{FR}=\frac{PD}{DO}.............(2)$
From equations (1) and (2), we have
$\frac{PE}{EQ} = \frac{PF}{FR }$
Thus, EF || QR. (Converse of the basic proportionality theorem)
Hence proved.
Answer:
Given: AB || PQ and AC || PR
To prove: BC || QR
Since, AB || PQ so we have
$\frac{OA}{AP}=\frac{OB}{BQ}............(1)$
Also, AC || PR
$\frac{OA}{AP}=\frac{OC}{CR}..............(2)$
From equations (1) and (2), we have
$\frac{OB}{BQ} = \frac{OC}{CR }$
Therefore, BC || QR. (converse basic proportionality theorem)
Hence proved.
Answer:
Let PQ be a line passing through the midpoint of line AB and parallel to line BC, intersecting line AC at point Q.
i.e. $PQ||BC$ and $AP=PB$ .
Using the basic proportionality theorem, we have
$\frac{AP}{PB}=\frac{AQ}{QC}.........(1)$
Since $AP=PB$
$\frac{AQ}{QC}=\frac{1}{1}$
$\Rightarrow AQ=QC$
$\therefore$ Q is the midpoint of AC.
Answer:
Let P is the midpoint of line AB, and Q is the midpoint of line AC.
PQ is the line joining midpoints P and Q of lines AB and AC, respectively.
i.e. $AQ=QC$ and $AP=PB$ .
We have,
$\frac{AP}{PB}=\frac{1}{1}..........................1$
$\frac{AQ}{QC}=\frac{1}{1}...................................2$
From equations (1) and (2), we get
$\frac{AQ}{QC}=\frac{AP}{PB}$
Therefore, by the basic proportionality theorem, we have $PQ||BC$
Answer:
Draw a line EF passing through point O such that $EO||CD\, \, and\, \, FO||CD$
To prove:
$\frac{AO}{BO} = \frac{CO}{DO}$
In $\triangle ADC$ , we have $CD||EO$
So, by using the basic proportionality theorem,
$\frac{AE}{ED}=\frac{AO}{OC}........................................1$
In $\triangle ABD$ , we have $AB||EO$
So, by using the basic proportionality theorem,
$\frac{DE}{EA}=\frac{OD}{BO}........................................2$
Using equations (1) and (2), we get
$\frac{AO}{OC}=\frac{BO}{OD}$
$\Rightarrow \frac{AO}{BO} = \frac{CO}{DO}$
Hence proved.
Answer:
Draw a line EF passing through point O such that $EO||AB$
Given :
$\frac{AO}{BO} = \frac{CO}{DO}$
In $\triangle ABD$ , we have $AB||EO$
So, by using the basic proportionality theorem,
$\frac{AE}{ED}=\frac{BO}{DO}............(1)$
However, it is given that
$\frac{AO}{CO} = \frac{BO}{DO}.............(2)$
Using equations (1) and (2), we get
$\frac{AE}{ED}=\frac{AO}{CO}$
$\Rightarrow EO||CD$ (By basic proportionality theorem)
$\Rightarrow AB||EO||CD$
$\Rightarrow AB||CD$
Therefore, ABCD is a trapezium.
Also Read-
1. Similar Figures: Two figures are said to be similar if their corresponding sides are in the same proportion and corresponding angles are equal.
2. Conditions for Similar Triangles: The criteria for similarity of the triangle are AA (Angle-Angle), in which two angles of one triangle are equal to the corresponding two angles of another triangle.
3. Midpoint Theorem: In this theorem, the line segment that joins the midpoint of the two sides of the triangle is parallel to the third side of the triangle, and the length of this line segment is half of the third side of the triangle.
4. Parallel Lines and Transversals: A transversal divides the sides of the triangle into proportional segments when it intersects the two parallel lines.
Also see-
Students must check the NCERT solutions for Class 10 Maths and Science given below:
Students must check the NCERT exemplar solutions for Class 10 Maths and Science given below:
Frequently Asked Questions (FAQs)
The ratio of the areas of the triangles is equal to the square of the ratio of the sides.
Ratio of area = 1:4
4/x=1/3
x=12cm
The length of the longer side is 12cm.
It will always be parallel to the third side of the triangle.
The congruent figures are all similar, but the opposite is not true.
If we know the side ratio and the length of any side, we can find the length of the corresponding side of the other triangle by using the ratio.
The ratio of the areas of the triangles is equal to the square of the ratio of the sides.
No, resemblance only relates to the shape of the figures and does not imply equality.
The term "similar" refers to two figures that have the same shape but differ in size.
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